An odd function of period 2$\pi$ is appoximated by a Fourier Series with N terms. The appoximate error as measured by mean-square deviation is
$$E_N =\int\limits_{-\pi}^\pi\left( f(x) – \sum_{n=1}^N b_n \sin nx \right)^2 dx$$
By differentiating $E_N$ with respect to the coefficients $b_n$, find the values of $b_n$ that minimize $E_N$.
The main problem for me was I didn't know how to differentiate this function. Any help will be appreciated. Thanks.
Best Answer
An idea only: putting $\,b:=(b_1,b_2,...,b_N)\,$ and regarding the given function as a real one on $\,\Bbb R^N\,$ :
$$\frac{d}{db}(E_N)=\frac{d}{db}\left[\int\limits_{-\pi}^\pi \left(f(x)-\sum_{n=1}^N b_n\sin nx\right)^2dx\right]=\sum_{k=1}^N\int\limits_{-\pi}^\pi\frac{d}{db_k}\left(f(x)-\sum_{n=1}^Nb_n\sin nx\right)^2dx$$
Assuming differentiation under the integral sign is allowed (Leibnitz theorem)